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Ratio

A ratio is a relationship between two or more quantities. It expresses the relative proportions of various quantities. A ratio is written with a colon ($$:$$), as in $$3:4$$.

A café makes biscuits using $$\Tblue{2}$$ kilos of flour and $$\Tred{1}$$ kilo of sugar. The ratio of flour to sugar is $$\Tblue{2}:\Tred{1}$$.

I make biscuits with $$\Tblue{8}$$ ounces of flour and $$\Tred{4}$$ ounces of sugar. The ratio flour to sugar is the same $$\Tblue{8}:\Tred{4} = \Tblue{2}:\Tred{1}$$, because I need twice as much flour as I need sugar.

Ratios can compare more than two quantities.

My recipe uses as much butter as it does sugar and twice as much flour. The ratio of flour to sugar to butter is $$\Tblue{2}:\Tred{1}:\Tviolet{1}$$.

A TV screen has a width of $$\Tblue{48}$$ cm and a height of $$\Tgreen{27}$$ cm. Its width to height ratio is $$\Tblue{48}:\Tgreen{27}= \Tblue{16}\times\Tred{3}:\Tgreen{9}\times\Tred{3} = \Tblue{16}:\Tgreen{9}.$$

This television has an aspect ratio of 16:9. This refers to the lengths of its sides. The ratio 16:9 is in its simplest form.

A ratio can be written in the form $$1:n$$ or $$n:1$$. The number $$n$$ can be any positive number, not just an integer. It is usually written in fractional or in decimal form.

The ratio $$\Tblue{3}:\Tgreen{7}$$ can be written as $$\Tblue{1}:\frac{\Tgreen{7}}{\Tblue{3}}$$ or in decimal form $$\Tblue{1}:\Tgreen{2.33}$$.

The form $$1:n$$ is mainly used to compare ratios.

An alloy has a ratio of $$\Tblue{\text{copper}}$$ to $$\Tgreen{\text{tin}}$$ of $$\Tblue{36}:\Tgreen{17}$$ and another has a ratio of $$\Tblue{20}:\Tgreen{13}$$. The second alloy has more $$\Tgreen{\text{tin}}$$ because $$\Tblue{36}:\Tgreen{17} = \Tblue{1}:\frac{\Tgreen{17}}{\Tblue{36}} = \Tblue{1}:\Tgreen{0.47},\qquad \Tblue{20}:\Tgreen{13} = \Tblue{1}:\frac{\Tgreen{13}}{\Tblue{20}} = \Tblue{1}:\Tgreen{0.65}.$$

Brass is an alloy of copper and zinc. Different types of brass use the metals in different ratios.

Finding one quantity from the other. In a class with a ratio of boys to girls of $$\Tblue{2}:\Tgreen{3}$$, there are $$\Tblue{12}$$ boys. The number of girls is $$ \frac{\Tblue{12}}{\Tblue{2}}\times\Tgreen{3}= 6\times\Tgreen{3} = \Tgreen{18}. $$

Splitting quantities. There are $$\Torange{45}$$ pupils in the class. The proportion of boys is $$\displaystyle\frac{\Tblue{2}}{\Tblue{2}+\Tgreen{3}} = \frac{\Tblue{2}}{\Torange{5}}$$. The number of boys is $$ \frac{\Torange{45}}{\Torange{5}}\times\Tblue{2} = 9\times\Tblue{2} = \Tblue{18}.$$

Ratio with more than two numbers. A sum of $$\Torange{20}$$ euros is to be split between John, Mark and Paul in a ratio $$\Tblue{2}:\Tgreen{7}:\Tviolet{1}$$. They will receive in euros \begin{align*} &\frac{\Torange{20}}{\Tblue{2}+\Tgreen{7}+\Tviolet{1}}\times\Tblue{2} = \frac{\Torange{20}}{\Torange{10}}\times\Tblue{2} = 20\times\Tblue{2} =\Tblue{40},\\ &\quad\frac{\Torange{20}}{\Torange{10}}\times\Tgreen{7} = \Tgreen{140},\quad \frac{\Torange{20}}{\Torange{10}}\times\Tviolet{1} = \Tviolet{20}. \end{align*}